arXiv:1709.06070v2 [math.RA] 21 Sep 2017
FRIEDRICH MARTIN SCHNEIDER AND JENS ZUMBR ¨AGEL
Abstract. Finite Frobenius rings have been characterized as precisely those finite rings satisfying the MacWilliams extension property, by work of Wood. In the present note we offer a generalization of this remarkable result to the realm of Artinian rings. Namely, we prove that a left Artinian ring has the left MacWilliams property if and only if it is left pseudo-injective and its finitary left socle embeds into the semisimple quotient. Providing a topological perspective on the MacWilliams property, we also show that the finitary left socle of a left Artinian ring embeds into the semisimple quotient if and only if it admits a finitarily left torsion-free character, if and only if the Pontryagin dual of the regular left module is almost monothetic. In conclusion, an Artinian ring has the MacWilliams property if and only if it is finitarily Frobenius, i.e., it is quasi-Frobenius and its finitary socle embeds into the semisimple quotient.
Introduction
Quasi-Frobenius rings, i.e., rings which are Artinian and self-injective, belong to those clas-sical Artinian rings which have been a driving force in the development of modern ring and module theory. Introduced by Nakayama [Nak41], one of their many equivalent characteriz-ations identifies them as those rings R for which the Hom(−, R) functor provides a duality between its finitely generated left modules and its finitely generated right modules. They also appear as the smallest categorical generalization of group rings of finite groups.
Within the class of quasi-Frobenius rings, those rings R for which the socle socR is iso-morphic, as one-sided module, to the semisimple quotient ring R/radR, are the Frobenius rings. They emerge naturally as a generalization of Frobenius algebras, i.e., finite-dimensional algebras over a field that admit a non-degenerate balanced bilinear pairing.
In more recent years, with the advent of ring-linear coding theory (see, for example, [H+94]), the interest in finite ring theory has increased. One of the striking results in this regard is the characterization, due to Wood [Woo99,Woo08], of finite Frobenius rings as pre-cisely those ringsRwhich satisfy the following MacWilliams extension property: every Ham-ming weight preserving isomorphism between left submodules ofRn extends to a monomial transformation, i.e., is of the form (xi)7→(xσiui) for a permutationσ∈Snand invertible ring elementsui ∈R. This property has been established by MacWilliams [Mac62] in the case of
finite fields and consolidates the notion of code equivalence. A further remarkable result is the observation by Honold [Hon01] that in the finite case Frobenius rings can be characterized as those rings R satisfying a one-sided condition socR ∼= R/radR, even without assuming the ring to be quasi-Frobenius.
In the present note we offer a generalization of Wood’s characterization of finite Frobenius rings as rings satisfying the MacWilliams extension property to the realm of general infinite
Date: 22nd September 2017.
Key words and phrases. Quasi-Frobenius ring; Code equivalence.
F.M.S. acknowledges funding of the Excellence Initiative by the German Federal and State Governments.
(Artinian) rings. We remark that the case of infinite Artin algebras has recently been treated by Iovanov [Iov16].
While each (two-sided) MacWilliams ring is necessarily quasi-Frobenius, it turns out that the Frobenius property is in general too strong to be deduced from satisfying MacWilli-ams’ extension theorem. We therefore weaken the Frobenius property to a criterion which we call finitarily Frobenius, which merely requires that the finitary socle embeds into the semisimple quotient (either as left or right module). Clearly, every Frobenius ring is also fi-nitarily Frobenius. In our main result, Theorem4.7, we show that a left Artinian ring satisfies the MacWilliams extension property for left modules if and only if it is left pseudo-injective and the finitary left socle embeds into the semisimple quotient. It follows that an Artinian ring is finitarily Frobenius if and only if it satisfies MacWilliams’ extension property.
Our proof method is reminiscent of Wood’s approach [Woo99] and relies on the existence of certain torsion-free characters on finitarily Frobenius rings, as well as results on Pontryagin duality of discrete and compact abelian groups. Along the way, we show for a left Artinian ring that the finitary left socle embeds into the semisimple quotient if and only if it admits a finitarily left torsion-free character, if and only if the Pontryagin dual of the regular left module is almost monothetic.
1. Frobenius rings and generalizations
We compile in this section a few notions from ring and module theory as needed in the present context and introduce the notion of a finitarily Frobenius ring. For a comprehensive account on the classical theory we refer to [Lam99,AF92], see also [Woo99,Hon01].
In the following, the termring will always mean unital ring. Recall that a ring is said to bequasi-Frobenius if it is left Artinian and left self-injective, i.e., injective as a left module. As it turns out, the properties left Artinian and left self-injective can each be replaced by its right counterparts, and the Artinian property by Noetherian (cf. [Lam99], Sec. 15).
We shall, for a (left or right) module M, denote by socM the sum of all its minimal submodules, by radM the intersection of all maximal ones and by topM := M/radM its “top quotient”. Accordingly, we denote by radR the Jacobson radical of a ring R, i.e., the intersection of all maximal left (right) ideals; also, let soc(RR) be its left socle, i.e., the sum of all minimal left ideals, and let soc(RR) be its analogously defined right socle. A crucial
notion for the present note is the Frobenius property.
Definition 1.1. A ring R is calledFrobenius if it is quasi-Frobenius and satisfies (i) soc(RR)∼=R(R/radR) and/or (ii) soc(RR)∼= (R/radR)R.
For quasi-Frobenius rings the conditions (i) and (ii) are actually equivalent. Indeed, it is worthwhile to recall how the properties of quasi-Frobenius and Frobenius may be ex-pressed with respect to the principal decomposition, as we outline briefly below (for de-tails, see [Lam99], Sec. 16 or [AF92], Sec. 31). Let R be a left or right Artinian ring and let S := R/radR be its semisimple quotient. Then there is a list of orthogonal primitive idempotentse1, . . . , en∈R such that
R=Re1⊕. . .⊕Ren and R=e1R⊕. . .⊕enR
are direct sums of indecomposable left and right modules, respectively, and, letting ei :=
ei+ radR∈S for all i∈ {1, . . . , n}, one has decompositions
into simple left and right modules, respectively. For alli, j∈ {1, . . . , n} there holds
Sei∼=Sej ⇐⇒ Rei ∼=Rej ⇐⇒ eiR∼=ejR ⇐⇒ eiS ∼=ejS ,
and we may assume that Re1, . . . , Rem (for some m ≤ n) form a complete set of
non-isomorphic representatives for allRei. Then Se1, . . . , Sem (and e1S, . . . , emS) form an irre-dundant set of representatives for all simple left (right) modules. We shall refer toe1, . . . , em
as a basic set of idempotents for the ring R. It is easy to see that top(Rei) ∼= Sei and
top(eiR)∼=eiS (considered as R-modules), in particular, the former are simple.
Now if the ring R is quasi-Frobenius then each of soc(Rei) and soc(eiR) is also simple.
In fact, the following characterization is valid, which actually corresponds to Nakayama’s original definition of quasi-Frobenius rings [Nak41].
Theorem 1.2. LetRbe a left or right Artinian ring with a basic set of idempotentse1, . . . , em. Then the ring R is quasi-Frobenius if and only if there is a permutation π ∈Sm such that
soc(Rei)∼= top(Reπi) and soc(eπiR)∼= top(eiR).
The permutation π ∈ Sm in Theorem 1.2 is referred to as the Nakayama permutation. Notice that for any fixedjthe numberµj of indecomposablesReiisomorphic toRejequals the number of simples top(Rei) isomorphic to top(Rej), and coincides with its right counterpart. Hence, for a quasi-Frobenius ring R, Theorem1.2 yields that
soc(RR) =
n
L
i=1
soc(Rei)∼=
n
L
i=1
top(Rei) = top(RR)
if and only ifµπi=µi for alli∈ {1, . . . , n}, which in turn is equivalent to soc(RR)∼= top(RR). This shows the equivalence of condition (i) and (ii) of Definition 1.1 for quasi-Frobenius rings. (On the other hand, any Artinian ring R satisfying both soc(RR) ∼= top(RR) and soc(RR)∼= top(RR) is necessarily quasi-Frobenius.)
We are going to introduce a finitary version of the Frobenius property. Given a ringR, we define its finitary left socle soc∗(RR) to be the sum of all finite minimal left ideals ofR, and
its finitary right socle soc∗(RR) as the sum of all finite minimal right ideals ofR.
Proposition 1.3. Let R be a quasi-Frobenius ring. Thensoc∗(RR) embeds into
R(R/radR) if and only if soc∗(RR) embeds into (R/radR)R.
Proof. Let e1, . . . , em be a basic set of idempotents for the ring R. First we observe that
soc(Rei) is finite if and only if top(eiR) is finite. Indeed, since R is quasi-Frobenius we have Hom(soc(Rei), R)∼= top(eiR) and Hom(top(eiR), R)∼= soc(Rei) (see [Lam99], Cor. 16.6 or [Woo99], Cor. 2.5). Furthermore, ifT is any finite simple module, then Hom(T, R) is finite, since every homomorphismT →R maps into the finite set soc∗(R).
Next it is easy to see that top(Rei)∼=Sei is finite if and only if eiS∼= top(eiR) is finite; in fact they are isomorphic to the standard column and row modules of the same matrix ring in the Artin-Wedderburn decomposition ofS :=R/radR. This shows that soc(Rei), top(Rei),
soc(eiR), top(eiR) are, for each i, simultaneously either finite or infinite.
Now denoting by F the set of all i such that soc(Rei) is finite, we see from Theorem 1.2
that the Nakayama permutation π preserves the set F, and from the subsequent discussion that soc∗(RR) embeds into top(RR) if and only ifµπi =µi for alli∈F, which holds if and
only if soc∗(R
In view of Proposition 1.3, we record the following definition.
Definition 1.4. A ringRis calledfinitarily Frobenius if it is quasi-Frobenius and there holds one (thus each) of the following equivalent conditions:
(i) soc∗(
RR) embeds intoR(R/radR).
(ii) soc∗(RR) embeds into (R/radR) R.
The following notion will also be relevant for the MacWilliams extension property. A left module RM is said to bepseudo-injective if for every submodule N of M and any injective homomorphism f:N → R there is a homomorphismg:M → R withg|N =f. Accordingly, a ring R is called left (right) pseudo-injective if for every left (right) ideal I, each injective homomorphism I →R is given by a right (left) multiplication by an element of R.
Clearly, every injective module is pseudo-injective and every quasi-Frobenius ring is left and right pseudo-injective. Conversely, we note the following result.
Proposition 1.5. An Artinian ring is quasi-Frobenius if and only if it is both left and right pseudo-injective.
Proof. Observe that pseudo-injectivity impliesmin-injectivity, i.e., that for every minimal left (right) ideal I each homomorphismI →R is given by right (left) multiplication. The result is then a direct consequence of [Har82, Thm. 13], see also [Iov16, Thm. 3.12].
The following useful observation is implicit in [Iov16, Cor. 3.5], and for the reader’s con-venience we include a direct argument based on work of Bass [Bas64].
Lemma 1.6. Let R be a left or right Artinian ring which is left pseudo-injective. If RM is a leftR-module andg, h:M →Rare homomorphisms such that kerg= kerh, then there exists a unitu∈R such that h(x) =g(x)u for allx∈M.
Proof. Consider the induced injective maps eg,eh:M/N → R, where N := kerg = kerh. LettingI := imeg we have an injective homomorphismf :=eh◦eg−1:I →R, thus by pseudo-injectivity there exists a∈ R with f(z) = za for all z ∈ I, which implies h(x) = g(x)a for all x ∈ M. Similarly, we find b ∈ R such that g(x) = h(x)b for all x ∈ M. Now since
R =abR+ (1−ab)R ⊆aR+ (1−ab)R and R/radR is semisimple, it follows from [Bas64, Lem 6.4] that there is a unit u ∈ R such that u = a+ (1−ab)r for some r ∈ R. Thus
g(x)u=g(x)a+g(x)(1−ab)r =g(x)a=h(x) for all x∈M, as desired.
2. Torsion-free characters
In this section we show that any Frobenius ring admits a left (resp., right) torsion-free character and that, similarly, every finitarily Frobenius ring admits a finitarily left (resp., right) torsion-free character, cf. Definition2.1. Let us start with a fairly general setting.
Definition 2.1. LetRbe a ring and letEbe an abelian group. A homomorphismχ:R→E
is said to be left torsion-free (resp., right torsion-free) if the subgroup kerχ contains no nonzero left (resp., right) ideals. The homomorphismχ is calledtorsion-free if it is both left-and right torsion-free. Furthermore, χ:R→E is said to befinitarily left torsion-free (resp.,
finitarily right torsion-free) if the subgroup kerχcontains no nonzero left (resp., right) ideals; andχ is called finitarily torsion-free if it is both left- and right torsion-free.
Lemma 2.2. Every division ring D admits a torsion-free homomorphism into Q/Z.
Proof. The Z-module Q/Z has the cogenerator property, i.e., for any abelian group X and every nonzero x ∈ X there is a homomorphism f:X → Q/Z with f(x) 6= 0 (cf. [Lam99], Lem. 4.7). In particular, there exists a nonzero homomorphismχ:D→ Q/Z, which clearly must be torsion-free as Ddoes not admit any non-trivial left or right ideals.
Given a ring R and some integer n≥1, we consider the matrix ring Mn(R) :=Rn×n and
thetrace map tr : Mn(R)→R, m7→Pni=1mii, which is anR-bimodule homomorphism.
Lemma 2.3. Let R be a ring and let E be an abelian group. If a homomorphism χ:R→E is left torsion-free (resp., right torsion-free), then so is χ◦tr : Mn(R)→E for every n≥1.
Proof. Suppose thatχ:R→Eis left torsion-free and letI be a left ideal inMn(R) contained in ker(χ◦tr). Since the trace map isR-linear, it follows that tr(I) is a left ideal inRcontained in kerχ, and therefore tr(I) = 0. We claim that I = 0. Let a= (aij) ∈I, and denoting by
eij ∈M
n(R) the elementary matrix with (eij)ij = 1 and (eij)kℓ = 0 if (k, ℓ)6= (i, j), we have
tr(eija) = tr(eijP
k,ℓakℓekℓ) = tr(Pℓajℓeiℓ) = aji. Since eija∈ I for all i, j and tr(I) = 0,
we conclude a= 0 as desired. The proof for the right torsion-free case is analogous.
Lemma 2.4. LetR1, . . . , Rnbe rings and let E be an abelian group. For any left torsion-free (resp., right torsion-free) homomorphisms χi:Ri→E (i∈ {1, . . . , n}), the homomorphism
R1×. . .×Rn→E, (r1, . . . , rn)7→ n
P
i=1 χi(ri)
is left torsion-free (resp., right torsion-free), too.
Proof. Let R:= R1×. . .×Rn and let χ:R→ E, (r1, . . . , rn) 7→Pni=1χi(ri). Suppose that
the χi: Ri → E are left torsion-free and let I be a left ideal in R contained in kerχ. For
i∈ {1, . . . , n}denote by πi:R→Ri the projection and letei∈R be the central idempotent with (ei)
i = 1 and (ei)j = 0 forj 6= i. Thenπi(eiI) is a left ideal in Ri contained in kerχi.
It follows that πi(eiI) = 0 and thus eiI = 0 for all i ∈ {1, . . . , n}, which implies I = 0 as
desired. The proof for the right torsion-free case is analogous.
Corollary 2.5. Every semisimple ring admits a torsion-free homomorphism into Q/Z. Proof. Thanks to the famous Artin-Wedderburn theorem, any semisimple ring is isomorphic to Mn1(D1) ×. . . ×Mnm(Dm) for suitable positive integers n1, . . . , nm and division rings
D1, . . . , Dm. Thus we may apply Lemma2.2, Lemma 2.3and Lemma2.4.
We are now ready to establish the first main result of the present note. Our proof utilizes Pontryagin duality for modules, which we recall in the appendix.
Theorem 2.6. Let R be a left Artinian ring. The following are equivalent:
(1) soc∗(RR) embeds into
R(R/radR),
(2) R admits a finitarily left torsion-free homomorphism into Q/Z,
(3) R admits a finitarily left torsion-free character.
Proof. (1) =⇒(2). Since S := R/radR is semisimple there is by Corollary 2.5a torsion-free homomorphismχ:S→Q/Z. By hypothesis we have an embeddingϕ: soc∗(RR)→RS. Now
sinceQ/Zis divisible, i.e., injective as aZ-module, there exists a homomorphismχ:R→Q/Z
be any nonzero finite left ideal of R. Then we find a minimal (nonzero) left ideal I0 of R
such that I0 ⊆I. AsI0 is finite,I0 ⊆soc∗(RR). It follows thatϕ(I0) is a nonzero submodule
ofRS, i.e.,ϕ(I0) is a nonzero left ideal of the ringS. Sinceχis left torsion-free, we have that
ϕ(I0) *kerχ and thus I0 * ker(χ◦ϕ). By choice of χ and I0, this implies that I * kerχ.
This shows that kerχ contains no nonzero finite left ideal. (2) =⇒(3). Since T∼=R/Z, this is obvious.
(3) =⇒(1). Let R be left Artinian. By applying the Artin-Wedderburn theorem we find a finite semisimple ring E and a semisimple ring U without non-trivial finite left modules, together with a surjective homomorphism h: R → E ×U with kerh = radR. Consider the projection p: E×U → E and let hE := p◦h, so that K := kerhE = h−1(U). Since kerh = radR and U has no non-trivial finite left modules, it is easy to see that KT = 0 for every minimal finite left idealT of R. We conclude that Ksoc∗(R) = 0.
Now suppose that χ:R → T is a finitarily left torsion-free homomorphism. For each
a∈A:= soc∗(R) we have just shown thatK⊆ker(aχ), whence there exists a uniquea.χ ∈Eb
such that a.χ ◦hE = aχ. Moreover, viewing E as a right R-module, the homomorphism
hE: RR → ER induces a homomorphism ϕ: RA → REb, a 7→ a.χ. Furthermore, as χ is
finitarily left torsion-free we deduce that ϕis injective: if a∈A\ {0}, then Ra*kerχ, i.e., there exists r ∈R such that 1 6=χ(ra) = (a.χ)(hE(r)), wherefore a.χ6= 1. This shows that
RA embeds into REb. Finally, since E is finite and semisimple, thus Frobenius, we have that EEb∼=EE by work of Wood [Woo99, Thm. 3.10], and henceREb∼=RE. Thus the composition
of embeddings RA → REb → EE → R(R/radR) provides an embedding of RA = soc∗(RR)
into R(R/radR), as desired.
Let us also add a direct argument for the implication (3) =⇒(2) of Theorem 2.6. Suppose for a left Artinian ring R a finitarily left torsion-free homomorphism χ:R → R/Z is given. Since F := soc∗(RR) is finite,χ(F) is a finite subgroup ofR/Z, thus contained in the torsion
subgroupQ/Z. By divisibility ofQ/Z, there exists a homomorphismχ∗:R→Q/Zsuch that
χ∗|F = χ|F. In particular, F ∩(kerχ∗) = F ∩(kerχ). In turn, χ∗ must be finitarily left
torsion-free, as every finite left ideal of R contains a minimal left ideal. By the method of proof, we have the following.
Corollary 2.7. Let R be a left Artinian ring. If soc(RR)=∼R(R/radR) (in particular, if R is Frobenius), then R admits a left torsion-free homomorphism into Q/Z.
3. Dual modules and almost monotheticity
This section offers a topological perspective on (finitarily) Frobenius rings, in terms of compact modules arising via Pontryagin duality. Let us start off with a simple characterization of torsion freeness of characters. For notation, see the appendix.
Lemma 3.1. Let R be a ring and letχ∈R. The following hold.b
(1) The character χ is left torsion-free (resp., right torsion-free) if and only ifχR (resp., Rχ) is dense in R.b
(2) The character χ is finitarily left torsion-free (resp., right torsion-free) if and only if χ is not contained in a finite-index closed proper submodule of RRb (resp., RR).b
Proof. Consider the closed submoduleB :=χR≤RRb and the corresponding left ideal
Thenχis left torsion-free if and only if ∆(B) ={0}, which, by PropositionA.2, is the case if and only ifB =Rb. This proves (1). In order to show (2), we infer from (∗) thatχis finitarily left torsion-free if and only if ∆(B) has no nonzero finite left sub-ideals, which, thanks to Proposition A.2 and Lemma A.3, just means that B is not contained in any finite-index proper closed submodules ofRbR. Of course, the latter is equivalent toχ not being contained
in any finite-index proper closed submodules of RRb , which readily completes the argument.
The other cases are proven analogously.
Let us continue with a useful abstract concept for compact modules.
Definition 3.2. Let R be a ring. A compact right R-module XR is said to bemonothetic
if there exists x ∈ X such that xR = X. A compact right R-module XR is called almost monothetic if every finite cover of XR by closed submodules contains X itself, i.e., for every finite setMof closed submodules ofXR we have
X=[M =⇒ X ∈ M.
Note that both monothetic and almost monothetic compact modules provide a generaliza-tion of cyclic finite modules. Utilizing the following combinatorial Lemma3.3, we will provide a simple characterization of almost monothetic compact modules in Proposition3.4.
Lemma 3.3 ([Pas71], Lem. 5.2; see also [Got94], Thm. 18). LetG be an abelian group. IfH is a finite cover of G by subgroups such that G6=SH \ {H} for every H ∈ H, then G/TH is finite.
Proposition 3.4. Let R be a ring. A compact right R-module XR is almost monothetic if and only if every finite cover of XR by finite-index closed submodules contains X itself.
Proof. The implication (=⇒) is obvious. In order to prove (⇐=), let M be a finite set of closed submodules of XR with X =SM. We wish to show that X ∈ M. Without loss of generality, we may assume that X6=SM \ {M} for every M ∈ M. Thanks to Lemma 3.3, each member ofMthen has finite index inXR, whence X∈ M by our hypothesis.
Corollary 3.5. LetRbe a ring and letXR be a compact rightR-module. IfXR is not covered by its finite-index closed proper submodules, then XR is almost monothetic.
We now return to Pontryagin duals of Artinian rings. The subsequent result characterizes the finitarily Frobenius rings in topological terms, in turn offering an approach to the proof of the general MacWilliams theorem.
Theorem 3.6. Let R be a left Artinian ring. Then soc∗(RR) embeds into
R(R/radR) if and only if RRb is almost monothetic.
Proof. (=⇒) This follows from Theorem2.6 and Lemma3.1(2) along with Corollary 3.5. (⇐=) Suppose that RRb is almost monothetic. Since R is left Artinian, the set L of all finite simple left ideals of R is finite. By Proposition A.2 and Lemma A.3, the finite set
M:={Γ(I)|I ∈ L}consists of closed proper submodules ofRRb . AsRRb is almost monothetic, there exists χ∈Rb with χ /∈SM, i.e., I *kerχ for every I ∈ L. Since every nonzero finite left ideal of R contains a member ofL, it follows thatχ is finitarily left torsion-free. Hence, soc∗(
The following lemma is the main reason for our interest in almost monothetic modules.
Lemma 3.7. Let R be a ring, XR and YR be compact right R-modules, where XR is almost monothetic. Let f1, . . . , fn, g1, . . . , gn:XR→YR be continuous homomorphisms such that
Proof. Since by Theorem A.4the linear span ofYb is dense in C(Y), and by continuity of the map C(Y)→C,h7→Pni=1R h◦(fi−gi)dµX, our hypothesis implies that
We finish this section with the observation that, by the method of proof of Theorem 3.6, we have the following.
Corollary 3.8. Let R be a left Artinian ring. Ifsoc(RR)=∼R(R/radR) (in particular, ifR is Frobenius), then RRb is monothetic.
Proof. This is an immediate consequence of Corollary 2.7and Lemma 3.1(1).
4. MacWilliams’ extension theorem for the Hamming weight
In this section we prove MacWilliams’ extension theorem for the Hamming weight on general Frobenius rings. Let us start off with some basic terminology. Let Gbe an abelian group. By a weight on G we mean any function from G to C. Given a weight w:G → C
Lemma 4.1. Let G be an abelian group. For every x∈G,
We proceed to rings. Our main focus will be on the MacWilliams property. Let R be a ring and consider its unitary group
Definition 4.3. A ring R is called left MacWilliams if, for every integer n ≥ 1 and any homomorphism ϕ:RM → RN between submodules M, N of RRn with w
H(ϕ(x)) = wH(x)
for allx∈M, there existσ∈Snandu∈U(R)nwithϕ= Φσ,u|NM. Analogously, a ringR will be calledright MacWilliams if, for every integern≥1 and any homomorphismϕ:MR→NR
between submodules M, N of RRn withwH(ϕ(x)) =wH(x) for all x ∈M, there existσ ∈Sn
andu∈U(R)n withϕ= Ψσ,u|N M.
Our goal is to establish a link between the MacWilliams property and the finitary Frobenius property. The next lemma constitutes a key observation.
Lemma 4.4. Let R be a ring such that RRb is almost monothetic. Let n≥1, let M be a left R-module and let ϕ, ψ: RM → RRn be homomorphisms with w
H(ϕ(x)) = wH(ψ(x)) for all x∈M. Then,
∀j∈ {1, . . . , n} ∃k∈ {1, . . . , n}: kerψk⊆kerϕj.
Proof. In light of Corollary4.2, our assumption means that
∀x∈M:
The result now follows by applying Lemma3.7withXR=RRb and YR=MRc , together with
Pontryagin duality.
Lemma 4.5. Let R be a ring such that RRb is almost monothetic. Let n≥1, let M be a left
We are ready to prove the generalized MacWilliams extension theorem.
Proposition 4.6. Every left Artinian, left pseudo-injective ringR such thatsoc∗(RR)embeds into R(R/radR) is left MacWilliams.
Proof. By virtue of Theorem 3.6 we have that RRb is almost monothetic. Given n≥ 1, any left R-module M and homomorphisms ϕ, ψ:RM → RRn with w
H(ϕ(x)) =wH(ψ(x)) for all x ∈ M, we need to show that there exist σ ∈ Sn and u ∈ U(R)n such that ψ = Φσ,u◦ϕ.
Our proof proceeds by induction on n≥ 1. For the induction base, let n = 1. Since RRb is almost monothetic, Lemma4.5implies that kerϕ= kerψ. Thanks to Lemma1.6, there exists
u∈U(R) such that ψ(x) =ϕ(x)u for all x∈ M, i.e., ψ= Φid,u◦ϕ. For the inductive step,
suppose that the statement is true for some n≥1. AsRbR is almost monothetic, Lemma 4.5
implies that there exist j, k ∈ {1, . . . , n+ 1} such that kerϕj = kerψk. Using Lemma 1.6
Now we present the main result, which characterizes the Artinian rings satisfying the MacWilliams extension property. In addition to the results of the previous sections, we use a strategy developed by Dinh, L´opez-Permouth [DL04] and Wood [Woo08].
Theorem 4.7. A left Artinian ring R is left MacWilliams if and only if it is left pseudo-injective and soc∗(R) embeds into
R(R/radR).
Proof. Every left Artinian, left pseudo-injective ring R with an embedding of soc∗(R) into R(R/radR) is left MacWilliams by Proposition 4.6. For the converse, suppose the ringR to
someSei. Also, there are natural numbersν1, . . . , νm with soc(RR)=∼Lmi=1(Sei)νi. Without
loss of generality, we may assume that the finite modules Sei are precisely Se1, . . . , Seℓ for
some ℓ ≤ m. We conclude that soc∗(RR) embeds into RS if and only if νi ≤ µi for every
i∈ {1, . . . , ℓ}. Now assuming for contradiction that soc∗(RR) does not embed intoRS, there
exists somej∈ {1, . . . , ℓ} such thatνj > µj.
AsRSej is isomorphic to the pull-back toRof the standard column module over Mµj(Dj),
we may assume that soc∗(RR) contains a matrix moduleA:=Dµj×νj
j over Mµj(Dj). Wood’s
result [Woo08, Thm. 4.1] states the existence of left modulesC+, C− ofAn for some positive
integer nand a Hamming weight preserving isomorphismf:C+→C− such that C+ has an
identically zero component while C− does not. Of course, C+, C− are submodules of RRn
and the isomorphism f: C+ → C− cannot be extended to a monomial transformation,
con-tradicting the left MacWilliams property.
We conclude this note with multiple characterizations of the MacWilliams extension prop-erty for quasi-Frobenius rings.
Corollary 4.8. Let R be a quasi-Frobenius ring. The following are equivalent:
(1) R is finitarily Frobenius,
(2) R admits a finitarily left torsion-free character,
(3) RRb is almost monothetic,
(4) R is left MacWilliams.
Also, each of (2) and (4) may be exchanged by its right version, and (3) by its left version.
Proof. We have (1)⇐⇒(2) by Theorem 2.6, (1)⇐⇒(3) is due to Theorem 3.6, and, noting thatR is pseudo-injective, (1)⇐⇒(4) according to Theorem 4.7. The last statement follows
from the symmetry of (1), see Proposition1.3.
Finally, let us state the following characterization of Artinian rings satisfying the MacWil-liams property on both sides.
Corollary 4.9. An Artinian ring is finitarily Frobenius iff it is left and right MacWilliams. Proof. This follows at once from Proposition1.5 and Theorem4.7.
Appendix A. Abstract harmonic analysis
In this appendix, we shortly recollect some basics of abstract harmonic analysis: Pontryagin duality, Bohr approximation, and Haar integration. For more on this, we refer to [DE09].
Consider the circle groupT:={z∈C| |z|= 1}, which is isomorphic to the quotient R/Z. Notice that the groupTis written multiplicatively, whereas the group R/Zadditively. LetG
be a locally compact abelian group. We denote byGb thedual group ofG, i.e., the topological group of all continuous homomorphisms from G into T endowed with the compact-open topology, which constitutes a locally compact abelian group itself. As usual, the elements ofGb are called characters onG.
Theorem A.1 (Pontryagin). If Gis a locally compact abelian group, then the map ηG:G→G,bb g7→(γ 7→γ(g))
Ifϕ:G→His a continuous homomorphism between locally compact abelian groups, then the map ϕb:Hb →Gb,β 7→ β◦ϕ defines again a continuous homomorphism. The assignment
G7→Gbbecomes this way a functor from the category of locally compact abelian groups (with continuous homomorphisms as morphisms) into itself, and Theorem A.1 actually provides a natural equivalence between a locally compact abelian group and its bidual.
We are going to recollect some bits about annihilating subgroups. Let G be a locally compact abelian group. For subsetsA⊆Gand B ⊆Gb, let us define
Γ(A) :={γ ∈Gb|γ|A≡1}, ∆(B) :=\{kerγ |γ ∈B},
noting that Γ(A) is a closed subgroup of Gb, while ∆(B) constitutes a closed subgroup ofG.
Proposition A.2. Let G be a locally compact abelian group. Then Γ and ∆ constitute mu-tually inverse order-reversing bijections between the closed subgroups of G and G. Moreover,b for any closed subgroups H ≤G and K≤G,b
Γ(H)∼=G/H,[ ∆(K)∼=G/K,[b
wherefore Γ(H)is finite if and only if H has finite index inG, and ∆(K) is finite if and only if K has finite index in G.b
Let us briefly turn to (discrete) rings and compact modules arising via Pontryagin duality. For a left (resp., right) R-module M, the compact dual group Mc := Hom(M,T) admits a continuous right (resp., left) R-module structure given by
(χr)(x) :=χ(rx), (resp., (rχ)(x) :=χ(xr)) χ∈M , rc ∈R, x∈M.
Moreover, ifϕ: M →N is a homomorphism between left (resp., right)R-modules M and N, then the continuous homomorphismϕb:Nb →Mc,χ7→χ◦ϕis in line with the right (resp., left)
R-module structure of Nb and Mc. In particular, this construction applies to theR-bimodule
M =RRR, which therefore gives rise to the compact R-bimoduleRRRb .
Lemma A.3. Let R be a ring. A subgroup A of a left (resp., right) R-module M is a submodule of M if and only if Γ(A) is a submodule of the right (resp., left) R-module Mc. In particular, a subgroup I ≤ R is a left (resp., right) ideal in R if and only if Γ(I) is a submodule ofRRb (resp., RR).b
Next we recall the Bohr approximation theorem. Given a compact (Hausdorff) space X, let us consider the commutativeC∗-algebra C(X) of all continuous complex-valued functions
onX, equipped with the obvious point-wise operations and the supremum norm.
Theorem A.4 (Bohr approximation theorem). Let G a compact abelian group. Then Gb generates a dense linear subspace of C(G).
Proof. By TheoremA.1,Gb separates the points ofG, whence the Stone-Weierstrass theorem asserts that the∗-algebraAgenerated byGbis dense inC(G). ButAcoincides with the linear subspace generated by Gb, simply because Gb is closed under point-wise complex conjugation
and multiplication. Hence, the theorem follows.
Lemma A.5. Let G be a compact abelian group. For every χ∈G,b
Z
χ dµG = (
1 if χ= 1,
0 otherwise.
Proof. Clearly, R 1dµG= 1. Suppose that χ6= 1. If g∈Gsuch that χ(g)6= 1, then Z
χ(x)dµG(x) = Z
χ(gx)dµG(x) = Z
χ(g)χ(x)dµG(x) =χ(g) Z
χ(x)dµG(x),
which implies that R χ dµG= 0. This completes the proof.
Acknowledgments
The authors would like to thank Tom Hanika for providing a creative and cheerful atmo-sphere during the final stage of this work.
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F.M.S., Institut f¨ur Algebra, TU Dresden, 01062 Dresden, Germany
E-mail address: martin.schneider@tu-dresden.de
J.Z., Faculty of Computer Science & Mathematics, Universit¨at Passau, 94032 Passau, Germany
